A conjecture on the behavior of tails of fixed points of the shot noise transform
Author:
O. M. Iksanov
Translated by:
Oleg Klesov
Journal:
Theor. Probability and Math. Statist. 69 (2004), 55-60
MSC (2000):
Primary 60E07; Secondary 40E05
DOI:
https://doi.org/10.1090/S0094-9000-05-00613-7
Published electronically:
February 7, 2005
MathSciNet review:
2110904
Full-text PDF Free Access
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Abstract: We show, under some restrictions on the response function depending on a parameter $\alpha$, that the tails of fixed points of transforms of a Poisson shot noise process are proportional at infinity to the exponential function of order $-\alpha$ if $\alpha \in (1,2)$. We advance an argument in support of the conjecture that this result remains true for $\alpha \geq 2$.
References
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- Aleksander M. Iksanov and Zbigniew J. Jurek, On fixed points of Poisson shot noise transforms, Adv. in Appl. Probab. 34 (2002), no. 4, 798–825. MR 1938943, DOI https://doi.org/10.1239/aap/1037990954
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References
- N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1989. MR 1015093 (90i:26003)
- A. K. Grinčevicjus, On a limit distribution for a random walk on a line, Lit. Mat. Sbornik 15 (1975), no. 4, 79–91. (Russian) MR 0448571 (56:6877)
- A. M. Iksanov, On perpetuities related to the size-biased distributions, Theory Stoch. Proc. 8(24) (2002), no. 1–2, 128–135. MR 2028745 (2004m:60146)
- A. M. Iksanov and Z. J. Jurek, On fixed points of Poisson shot noise transforms, Adv. Appl. Prob. 34 (2002), 798–825. MR 1938943 (2003i:60021)
- K. S. Lau and C. R. Rao, Solution to the integrated Cauchy functional equation on the whole line, Sankhyā A46 (1984), 311–318. MR 0798038 (86m:39017)
- Q. Liu, On generalized multiplicative cascades, Stoch. Proc. Appl. 86 (2000), 263–286. MR 1741808 (2001b:60102)
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Additional Information
O. M. Iksanov
Affiliation:
Faculty for Cybernetics, Kyiv Taras Shevchenko National University, Kyiv 01033, Ukraine
Email:
iksan@unicyb.kiev.ua
Received by editor(s):
November 14, 2002
Published electronically:
February 7, 2005
Article copyright:
© Copyright 2005
American Mathematical Society
